BOSTON UNI'/ERSITY
SCHOOL OF EDUCATION
ANALYSIS OF DENOMINATE NUMBERS IN TWO SERIES
OF ARITHMETIC TEXTBOOKS
Submitted by
Henry Arnold Schulz
In oartial fulfilment of requirements for the
degree of Master of Education
1933
Bosion University
School of Education
Library
I
Table of Contents
Introduction Pa,
The Study-
Part I Strayer-Upton Arithmetics
Relations among units of measure
Informational matter with reference
to units of measure
Units of measure merely as names
Table facts in examoles and problems
Reductions in denominate numbers
Computation with denominate numbers
Summary
Part II Hoyt-Peet Arithmetics
Relations among units of measure
Informational matter with reference
to units of measure
Units of measure merely as names
Table facts in examples and nroblems
Reductions in denominate numbers
Computation with denominate numbers
Summary
Final Summary and Concluding Observations
Bibliography
INTRODUCTION
This study reports the analysis of two arithmetic
text-book series with respect to the units of weight
and measure and more especially the denominate numbers,
that is, numbers which refer to units of measure, as,
for example, 3 inches, 4 pounds. Miss Vernette B.
Moore constructed an analysis of denominate numbers
found in four text-books for the purpose of determining
the concepts and skills needed in teaching denominate
1
numbers. This study assumed that the elements and
processes found in these text-books were proper teaching
material. The studies by Wilson with respect to the
social usage of arithmetic and the more recent
2 3 4
investigations of Louth, Sala, and Traniello, into the
uses of weights and measures in business and industry,
present evidence which leads us to be critical of the
subject matter found in text-books in connection with
denominate numbers. The first step in a critical
1. Moore, Vernette B» An Analysis of Denominate
Numbers with Data on Their Teaching. State
University of Iowa, 1929,
2. Louth, Mary. Units of Measurement in Industry.
B.U. Graduate School thesis, 1931.
3. Sala, V. Denominate Numbers used in the Factories of
New Britain* B.U. School of Education thesis, 1931.
4» Traniello, A. Use of Tables of Measurement in
Business. B.U. School of Education thesis, 1930.
2
examination of a text-book in respect to denominate
numbers must necessarily be an analysis which shows
the content of the text-book*
The present study offers an analysis of each of
the following two text-book series:
Strayer-Upton
Strayer-Upton Arithmetics
American Book Company
Three Book Series^ 1928 edition
Hoyt and Peet
The New Everyday Arithmetic
Houghton Mifflin Company
Three Book Series; first book, 1930 edition;
second and third book, 1927 edition.
Units of weight and measure have been classified
under six main headings, namely,
1» Relations among units of measure.
2. - Informational matter with reference to
weights and measures.
3. Denominate numbers merely mentioned in
written problems and involving no computation
with denominate numbers as such.
4. Table facts in examples and written problems.
5. Reduction of denominate numbers*
6. Computation with denominate numbers.
The units of weight and measure assembled under
each of these main headings have been further sub-divided
into groups according to the usual •^table" classification*
3
In other words, all units of measure dealing with length
have been grouped under "Linear Measure"; those
dealing with square units of measure, under "Square
Measure", and so on.
This analysis does not include tables of weights
and measures found in the rear of the text'-books. These
tables are intended for reference and pupils are not required
to memorize them as they are in the case of those tables
found in the text proper. All computation with
United States money (including changing of money),
computation with metric measures, as well as computations
by the 100, or by the 1000, have been excluded from
the analysis, since they are decimal. Examples and
written problems in mensuration have also been excluded,
since they properly form a part of intuitive geometry,
rather than of denominate numbers. It also appears
proper to exclude work in connection with telling time,
making schedules for buses, trolley cars and trains,
the use of the calendar, drawing to scale, graphs, work
interest, banking, commission, discount, installment
buying, stocks, bonds, insurance, equations, taxes, duty,
changing Fahrenheit readings to Centigrade readings and
visa versa, food problems involving the use of the terra
calories^
#
( ( I I <
#
PART ONE
The following analysis covers the units of weight and
measure, with special reference to denominate numbers,
as found in the three book series of the Strayer-Upton
Arithmetics.
r
4
RELATIONS AMONG UNITS OF MEASURE
Tables I to XIV which follow show the various
tables of measures found in the Strayer-Upton series
of arithmetics, each Table representing one table of
measure, except that counting articles and counting
paper have been grouped together under Table VIII and the
metric tables of length, of capacity, and of weight,
have been grouped together in Table XII* In each
case the equalities found in the table of measure has
been stated to the left of the figure and the frequency
of occurrence of these equalities is indicated in the upper
left corner of each square.. All other relations found in
the texts of the Strayer-Upton Series are listed to the
right and the frequency of occurrence of these equalities,
either in written problems or in some other isolated form,
is recorded in the lower right of the cell.
r
5
TABLE I- RELATIONS A..10NG UNITS OF UNITED STATES MONEY
FOUND IN THE STRAYER*UPTON SERIES.
Equalities to the left are found in tables; frequency of
occurrence is shown in upper left of each cell. All other
equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
10 mills s 1 cent
5 cents 1 nickel
lo cents St 1 dime
25 cents = 1 quarter
50 cents = § dollar
100 cents = 1 dollar
10 dimes = 1 dollar
Total
1 mill r 1/10 cent
1 nickel = 5 cents
1 dime = 10 cents
1 quarter = 25 cents
h dollar s 50 cents
1 dollar = 100 cents
1 dollar = 10 dimes
c
6
TABLE II. RELATIONS AMONG UNITS IN LINEAR MEASURE
FOUND IN THE 3TRAYER-UPT0N SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell*
All other equalities found in the texts are listed to the
right; frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence,.
12 inches
= 1 foot
4
1
f oot
5 12 inches
36 inches
= 1 yard
1
1
1
yard
- 36 inches
3 feet r 1
yard
3
1
yard
- 3 feet
16| feet =
1 rod
2
1
1
rod =
16| feet
5280 feet
5 1 mile
2
I
1
mile
= 5280 feet
5f yards =
1 rod
2
1
1
rod =
5f yards
5
1
mile
= 1760 yards
320 rods -
1 mile
2
1
mile
- 320 rods
Total
16
7
c
t
r
TABLE III. RELATIONS AMONG UNITS IN SQUARE MEASURE
FOUND IN THE STRAYER-UPTON SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell*
Blank indicates no occurrence*.
144 sq»in* « 1 sq»rt^
9 sq,.ft» r 1 sq..yd*
30f sq.yd» - 1 sq,.rd*
160 sq^rd^ - 1 acre
43,560 sq»ft^ - 1 acre
640 acres ^ 1 sq..mi
Total
20
1 sq# ft.. — 144 sq#in.
1 sq» yd» = 9' sq» ft*
1 sq» rd^ = 30t sq» yd,
1 acre = 160 sq» rd,
1 acre s 43,560 sq* ft,
1 sq.. mi • X 640 acres
8
TABLE IV.
RELATIONS AMONG UNITS IN CUBIC MEASURE
FOUND IN THE 3TRAYER-UPT0N SERIES^
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the
right; frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
3
1728 cu,in, K 1 cu,ft^
1 cu,. ft* s. 1728 cu, in.
3
27 cu,ft, = 1 cu,yd.
1 cu. yd, - 27 cu, ft.
6
Total
i
9
TABLE V. RELATIONS AMONG UNITS IN DRY MEASURE
FOUND IN THE STRAYER-UPTON SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
2 pt. 3 1 qt.
8 qt» = 1 pk.
4 pk. = 1 bu.^
Total..
1 qt. r 2 pt.
1 pk. t 8 qt.
1 bu. - 4 pk^
c
10
TABLE VI.
RELATIONS AMONG UNITS IN LIQUID MEASURE
FOUND IN THE STRAYER-UPTON SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower ri^ht of cell.
Blank indicates no occurrence.
3
2 pt^
1 qt»
1 qt.
2 pt.
3
4 qt,.
1 Kal»
1 gal.
4 qt,.
6
Total
II
TABLE VII. RELATIONS AL^ONG UNITS IN MEASURE OF WEIGHT
FOUND IN THE STRAYER-UPTON SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower rip;ht of cell*
Blank indicates no occurrence*
16 oz. = 1 lb*
3
1
1 lb, - 16 oz.
100 lb, - 1 cwt.
8
1 cwt, ^ 100 lb.
8000 lb, « 1 T.
3
2000 lb, =: 1 T,
Total
8
1
18
TABLE VIII. RELATIONS AMONG UNITS IN TABLES OF COUNTING
FOUND IN THE STRAYER-UPTON SERIES ►
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
12 units 9» 1 doz.
12 dozen = 1 gross
144 units T. 1 gross
20 units s 1 score
Total
24 sheets -s. 1 quire
480 (500) sheets * 1 ream
20 quires =. 1 ream
Total
1 doz. = 12 units
1 gross s 12 dozen
1 gross s 144 units
1 score s 20 units
1 quire = 24 sheets
1 ream =. 480 (500) sheets
1 ream - 20 quires
c
13
TABLE IX ► RELATIONS AMONG UNITS IN MEASURE OF TIME
FOUND IN THE STRAYER-UPTON SERIES*
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence..
60 seconds r 1 minute
60 minutes ^ 1 hour
24 hours =. 1 day
7 days r 1 week
365 days - 1 year
366 days = 1 leap year
4 weeks = 1 month
52 weeks s. 1 year
12 months - 1 year
100 years r 1 century
Total
28
1 minute = 60 seconds
1 hour = 60 minutes
1 day s 24 hours
1 week = 7 days
1 year =365 days
1 leap year 366 days
1 month = 4 weeks
1 year = 52 weeks
1 year = 12 months
1 century =. 100 years
c
14
TABLE X. RELATIONS AI/iCNG UNITS IN MEASURE OF CAPACITY
FOUND IN THE STRAYSR-UPTON SERIES •
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
1 gal, - 231 cu.in,
31| gal. = 1 bbl.
7| gal. s 1 cu.ft.
1 cu. ft. ^ 7^ gal.
Total
3
1
1
1
3
3
10
S
231 cu. in. c 1 gal.
1 bbl. X 3lf gal.
i
15
TABLE XI * RELATIONS AMONG UNITS IN MEASURE OF CONTENT
FOUND IN THE STRAYER-UPTCN SERIES »
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
1 bu, ^ 2150 cu. in.
8 bu,. » 1 cu, ft.
1 bu, s li cu, ft.
1 T, hard coal s 35 cu,f t
1 cu, ft. = 8 bu.
Total
12
2150 cu. in, r 1 bu.
It cu. ft, = 1 bu.
#
16
TABLE XI I • RELATIONS AMONG UNITS OF METRIC MEASURES
FOUND IN THE STRAY SR-UPTON SERIES.
Equalities to the left are found in tables. Frequency of
occurrence of these table facts is noted in upper left of cell.
1 mm. t 1/1000 m,
1 cm, s 1/100 m.
1 dm. - 1/10 ttt,
1 Dm, - 10 m.
1 Hm, = 100 m.
1 Km, - 1000 m.
Total
1 ml. = 1/1000 1,
1 HI. = 100 1,
Total
1 Kg, - 1000 g.
1 M. T. = 1000 Kg,
Total
c
17
TABLE XIII. RELATIONS BETWEEN UNITS OF METRIC MEASURES
AMD ENGLISH MEASURES FOUND IN STRAYER-UPTON
Equalities to the left are found in tables of measures*.
Frequency of occurrence of these table facts is noted in
upper left of cell. All other equalities found in texts
are listed to the right; frequency of occurrence is noted
in lower right of cell. Blank indicates no occurrence.
1 m. = 39,37 in.
1 tn, - 1.1 yd.
1 Km* tt ,6 mi.
Total
Total
1 liter = about 1 qt.
Total
1 Kg, r 2»2 pounds
18
TABLE XIV. RELATIONS IN MISCELLANEOUS MEASURES
FOUND IN THE STRAYER-UPTON SERIES ►
This table shows relations in miscellaneous measures
and the frequency of their occurrence in the Strayer-Upton
Arithmetics*
Money
1 English pound sterling = $4»8665 1
Linear Measure
1 knot 5 1.15 mi. I
Capacity and Content
1 cu.ft. ice s 56.4 lb, 1
1 cu.ft, water r 62.5 lb. 4
1 cu.ft. marble or granite - 168 lb» 2
1 cu.ft. pine = 25 1 lb. t
1 cu.ft. hard coal •= 57 lb. 1
1 bu.. potatoes^ 60 lb, 1
I bu. cherries r 50 lb» 1
I bu. wheat <=. 60 lb. I
1 bu. wheat :s 60 lb. 1
1 gal. waters 8.3 Ib^ I
Weight
1 lb. sugar = 2 cups 2
1 lb. flour s 4 cups I
1 pt. milk s 2 cups 1
1 lb. butter s 32 tbsp. 1
1 long ton = 2240 lb. I
1 ton of hay = 500 cu» ft. 1
Liquid
1 pt. = 2 glasses 1
1 qt. = 4 glasses 1
1 gal. s 16 glasses I
1
19
INFORMATIONAL MATTER WITH REFERENCE TO UNITS OF MEASURES.
Since actual use of units of weight and measure is always in
connection with articles and commodities, an understanding of
the meaning and use of units of weight and measure can only come
through commodities. As soon as a commodity is thoroughly un-
derstood, the appropriate unit of measure to be used in con-
nection with it has been learned incidentally.
Units of weight and measure are not learned as such, but
in connection with commodities. It is this idea of the con-
nection or inter-relation between commodities and units of
weight and measure that the expression "informational matter",
used here in connection with denominate numbers, is meant to
convey. No manipulation or computation with denominate
numbers is involved, but simply an understanding that a given
unit of measure is used in connection with a given commodity.
The instances in which nothing more is involved than an
understanding of the meaning of a unit of measure through its
use with a commodity, have been gathered and grouped under
two headings, namely, those appearing in statement form and
those appearing in question form. For instance, the
statement is made that "When we want to know how hot or how
cold it is, we look at a thermometer and read the number of
degrees of temperature."^ The information is imparted that
!• Strayer-Upton Arithmetics, Lower Grades, p. 238
(
20
temperature is measured by degrees and is listed under
"Informational Matter in Statement Form", Again, the
question is asked, "By what measures of weight are these thing
sold: Butter, spices, coffee, meat, iron, large quantities of
ice, small quantities of ice, pepper, hay?"^^ This is placed
under the heading of "Informational Matter in Question Form",-
Inf ormational Matter in Statement Form found in
Strayer-Upton..
Dry Measure:
Fruits, grains, vegetables are measured by the
pint, the quart, the peck, and the bushel.
Also sold by the pound..
Liquid Measurer
Milk is bought by the pint and quart;
Milk is bought by the gallon in large quantities;
Gasoline is bought by the gallon..
Measures of weight;
Ice is bought by the pound;
Ice is bought by the ton (large quantities);
Spices and pepper by the ounce;
Butter and meat by the pound;
Freight and express by the hundredweight;
Hundredweight is used in baying grain and cotton;
Hundredweight is used in selling milk to creamery;
Ton is used to weigh such things as coal and ice.
1- Strayer-Upton Arithmetics, Lower Grades, p# 237 #2
21
Miscellaneous Measuresr
Electricity
Gas
Lumber
Finishing lumber, molding
Temperature
Angles
Freezing point of water
Boiling point of water
Blood heat
kilowatt hour (watt hour)
cubic feet
board feet
lineal foot
Degree
degree
32 degrees F^
212 degrees F.
98i degrees F..
Informational Matter in Question Form found in
Strayer-Upton*
Telling ages of children, finding heights and weights*
What groceries are bought by the pound?
How are the following sold: butter, spices, coffee,
meat, iron, large quantities of ice, small
quantities of ice, pepper, hay?
Tell what unit of measure you use to findr
The height of a door;
The length of a carpet;
The length of a house;
The thickness of a board;
The weight of a ship;
The length of a road;
The weight of a man;
The length of a candle;
The amount of coal you burn each winter;
The amount of ice cream needed for 8 people;
The distance from Chicago to San Francisco;
The amount of gasoline used on an automobile trip.
o
22
UNITS OF MEASURE MERELY A3 NAMES
Many of the written problems contain units of measure which,
however, involve no computation or manipulation with denominate
numbers or such» The units of measure are mentioned merely as
words. The following problems mi,a;ht be cited to show the type
of written problem involved: "Express the following lengths to
the nearest tenth of a foot: 5.29 ft., 7»21 ft*, 4.38 ft.,
6»42 ft»"^ "How much must Mrs. King pay at the fish store for
some codfish weighing 2 5/3 lb. and costing 19 a pound."
"The Woolworth Building in the city of New York is 792 ft* high.
Jack's house is 72 ft. high. How many times as high as Jack's
3
house is the Woolworth Building?" "Mrs. Lee bought 3t yd. of
curtain goods. How many curtains, each 7/8 yd. long, can she
make from this?"^
1 Strayer-Upton, Middle Grades, p.216,j^4
2 Ibid* 88 #Q
3 " 32 #2
4 " 212/^5
23
TABLE XV. UNITS OF MEASURE AS '.VORDS IN STRAYER-UPTON
The table shows the frequency of occurrence of units of
measure involving no computation with denominate numbers
as such*
Linear Measure
Inch
Feet
Yard
Mile
7
16
63
22
Square Measure
Square yard
Acre
L
16
Cubic Measure
Cubic foot
1
Dry Measure
Quart
Bushel
3
5
Liquid Measure
Pint
Quart
Gallon
3
8
8
Wei^?ht Measure
Ounce
Pound
Ton
4
75
12
Counting
Pair
Dozen
24
37
24
Time Measure
Seconds I
Minutes 1
Hour 39
Day 32
Week 42
Month 25
Year 44
Miscellaneous Measures
Kilowatt hour
Cubic foot (gas)
5
1
25
TABLE FACTS IN EXAMPLES AND V/RITTSN PROBLEMS
Tables XVI to XXI show the number of times a table
fact or the reverse statement is called for in either
examples or in written problems. For example, a table
fact may be called for in an example of the following
1
type: "How many square inches equal 1 square foot?"
Or a written problem may involve the knowledge of a
table fact so that it may be solved, as for instance
in the following written problem: "Mr. Archer finds
that the distance around the deck of a steamship is 264
ft. How many times must he go around the deck to walk
2
a mile?" To solve this problem involves a knowledge
of the fact that 52SO feet equal i mile.
1. Strayer-Upton , Lower Grades, p. 305 #1,
8. Strayer-Upton, Middle Grades, p. 309 j^4.
Q
26
TABLE XVI. TABLE FACTS IN UNITED STATH:S MONEY ASKED IN
EXAMPLES AND WRITTEN PROBLEMS- STRAYER-UPTON .
In each instance, the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 10 mills equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: What is one nickel equal to?
Frequency of occurrence is shown in the column of cells
to the right. The frequency of occurrence in examples
is shown in the upper left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
10 millssCl cent)
5 cents=(l nickel)
10 centsnCl dime)
25 centssCl quarter)
50 centssCI dollar)
100 cents^Cl dollar)
10 dimeszCl dollar)
1 mill=(l/10 cent)
1 nickels (5 cents)
1 dimesClO cents)
1 quarter=(25 cents)
|- dollars(50 cents)
1 dollarsClOO cents)
1 dollarrClO dimes)
0
27
TABLE XVII TABLE FACTS IN LINEAR MEASURE ASKED IN EXAMPLES
AND WRITTEN PROBLEMS- STRAYER-UPTON
In each instance the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 12 inches equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: What is one foot equal to?
Frequency of occurrence is shown in the column of cells
to the right. The frequency of occurrence in examples
is shown in the upner left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
1 foot=(l2 inches)
1 yards (76 inches)
1 yard=(3 feet)
1 rod=(16| feet)
1 mile=(5280 feet)
1 rode (51 yards)
1 mile=(1760 yards)
1 milea(320 rods)
12 inchesaCl foot)
36 inchesrCl yard)
3 feet-(l yard)
16| feet=.(l rod)
5280 feetrCl mile)
5|- yards^Cl rod)
320 rodsrCl mile)
5
6
28
TABLE XVIII. TABLE FACTS IN SQUARE MEASURE ASKED IN EXAMPLES
AND '.VRITTEN PROBLEMS- STRAYER-UPTON
In each instance the fact in parenthesis indicates
the fact asked for. Read to the left of the fip^ure thus:
What are 144 square inches equal to? Frequency of occurrence
is shown in the column of cells to the left. Read to the
right of the figure thus: What is 1 square foot equal to?
Frequency of occurrence is shown in the column of cells to
the right. The frequency os occurrence in examples is
shown in the upper left of the cell; the frequency of
occurrence in written problems is shown in the lower right
of each cell.
144 sq. in.sd sq.ft.)
9 sq.ft.-sCl sq.yd.)
307 sq.yd.rd sq.rd.)
160 sq.rd. r(l A. )
640 A.-(l sq.mi.)
36 sq . mi . = (1 tp. )
1 sq.ft.s(144 square in.)
1 sq.yd. s(9 sq.ft.)
1 sq.rd. r(30t sq.yd.)
1 A.- (i60 sq.rd.)
1 sq.mi.T;(640 A. )
1 tp. =( 36 sq. mi . )
5
0
29
TABLE XIX. TABLE FACTS IN DRY MEASURE ASKED IN EXAMPLES
AND 'vVRITTEN PROBLEMS- STRAYER -UPTON
In each instance the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are S pints equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: How many pints are there in
one quart? Frequency of occurrence is shown in the
column of cells to the right. The frequency of occurrence
in examples is shown in the upper left of the cell; the
frequency of occurrence in written problems is shown
in the lower right of each cell. Blank indicates no
occurrence •
2 pintsaCl quart)
8 quarts^d peck)
4 pecks-id bushel)
1 quart=.(2 pints)
1 peck?:(8 quarts)
1 bushelTi(4 pecks)
30
TABLE XX. TABLE FACTS IN LIQUID MEASURE ASKED IN EXAMPLES
AND WRITTEN PROBLEMS — STRAYER-UPTON
In each instance the fact in parenthesis indicates the
fact asked for. Read to the left of the figure thus:
What are 2 pints equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus; One quart is equal to how many
pints? Frequency of occurrence is shown in the column of
cells to the right. The frequency of occurrence in
examples is shown in the upper left of the cell; the
frequency of occurrence in written problems is shown
in the lower right of each cell, ^lank indicates no
occurrence •
2 pints:t(l quart)
4 quarts^Cl gallon)
1 quart=(2 pints)
1 gallon=(4 quarts)
31
TABLS XXI. TABLE FACTS IN TIME ASKED IN EXAMPLES AND
'JVRITTEN PROBLEMS - STRAY^R-UPTON
In each instance the fact in parenthesis indicates the
fact asked for. Read to the left of the figure thus:What
are 60 seconds equal to? Frequency of occurrence is shown
in the column of cells to the left. Read to the right of
the figure thus: How many seconds in 1 minute? Frequency
of occurrence is shown in the column of cells to the right.
The frequency of occurrence in examples is shown in the
upper left of the cell; the frequency of occurrence in
written problems is shown in the lower right of each cell.
Blank indicates no occurrence*
60 secondssCl minute)
60 minutessd hour)
24 hourssCl day)
7 days::(l week)
365 dayssCl year)
366 dayszCl Ip.year)
52 weeks-id year)
12 monthssd year)
100 yearsrCl century)
1 minute:5(60 seconds)
1 hour=(60 minutes)
1 day-(24 hours)
1 weeks:.(7 days)
1 yearr(365 days)
1 year=(366 days)
1 years(52 weeks)
1 yearrd2 months)
1 century=.(lOO years)
32
REDUCTION OF DENOMINATE NUMBERS
Reduction of denominate numbers is the process of
changing their denomination without changing their value*
Denominate numbers may be changed from a higher to a
lower denomination, as for example, when 8 miles are
reduced to feet; denominate numbers may also be
changed from a lower to a higher denomination, as when
21 pints are reduced to gallons. The former process, is
known as descending reduction; the latter, as ascending
reduction.
Meaning and Reading of Tables XXII to XXXIII.
The upper figure shows descending reductions. The
unit of measure at the head of the column of vertical
squares in these figures read in connection with each
of the cells in these columns indicate the denomination
from which the reduction is made; when these terms are
read in connection with the row of horizontal cells to
the left, they indicate the denomination to which the
reduction is made,- The lower figures show ascending
reductions. The unit of measure read in connection with
the horizontal rows of cells to the right indicate the
denomi nation from which the reduction is made; read in
0
connection with the vertical column of cells it refers
to the unit to which the ascending reduction is made.
Frequency of occurrence of reductions appearing in
examples is shown in the upper left of each cell; those
appearing in written problems are shown in the lower
right of each cell.
34
TABLE XXII. REDUCTIONS IN UNITED STATES MONEY
STRAYEH-UPTON
The upper figure shows descending reductions: the lower
figure ascenaing reductions. The frequency of reductions
ap-earing in examples is shown in the upper left of cell;
those appearing in written problems in lower right. For
further details as to meaning and reading of table see page 32.
Dollars
Half dollars
Quarters
1
Dimes
1
1
Nickels
1
2
2
7
1
^ Cents
3
Dollars
Half dollars
Quarters
1
Dimes
1
Nickels
1
1
Cents 2
5
1
4
f
€
35
TABLE XXIII. REDUCTIONS IN LINEAR MEASURE-«-3TRAYER-UPT0N
The upper figure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each square; those appearing in written
problems are shown in the lower right of each square.
For further details as to meaning and reading of tables
see page 32^
Miles
2
Rods
2
1
Yards
6
1
9
3
Feet
1.
3
6
2
inches
Miles
Rods
Yards
8
Feet
Inches 1
;4
36
TABLS XXIV. REDUCTIONS IN SQUARE MEASURE^-tSTRAYSR-UPTON
The upper figure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each square; those appearing in written problems
are shown in the lower right of each square. For further
details as to meaning and reading of table see page 32,
Square *"iles
1
Acres
il
I t
Square Rods
Square Yards
Square Feet
Square inches
Square Miles
Acres
1
Square Rd.
Square Yards
Square Feet
2
1
Square inches
2
3
37
TABLE XXV. REDUCTIONS IN CUBIC MEASURE- STRAYER-UPTON
The upper fip;ure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the
upper left of each cell; those appearing in written
problems are shown in the lower right of each cell.
Cords
Cubic Yards
2
Cubic Feet
2
Cubic Inches
Cords
Cjabic Yards
3
Cubic Feet
6
38
TABLE XXVI. REDUCTIONS IN DRY MEASURE; STRAYER-UPTON
The upper figure shows descending reductions; the
lower figure af?cending reductions. The frequency of
reductions appearing in ex-mples is shown in the upper
left of each square; those appearing in written problems
are shown in the lower right of each square. For further
details as to reading and meaning of table see page 32.
Bushels
1
2
Pecks
6
4
5
2
Quarts
1
1
Pints
Bushels
Pecks
1 '
Quarts
1
4
2
1
Pints
3
1
39
TABLE XXVII. REDUCTIONS IN LIQUID MEASURE- STRAYER-UPTON
The upper figure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is snown in the upper
left of each cell; those appearing in written problems
are shown in the lower right of each cell. For further
details as to reading and meaning of table see page 32.
Barrels
Gallons
1
9
2
Quarts
9
1
4
1
Pints
2
2
Cups or Glasses
Barrels
Gallons
Pints
Quarts
|2~~
t
2 :
Cups or Glasses
4
40
TABLE XXVIII. REDUCTIONS IN WEIGHT MEASURES- STRAYER-UPTON.
The upper figure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions apnearing in examples is shown in the upper
left of each square; Those appearing in written problems
are shown in the lower right of each square. For further
details as to reading and meaning of table see page; 32,
Tons
Cwt.
4
1
2
Pounds
27
1
Ounces
Tons
Cwt.
1
Pounds
8
2
9
Ounces
20
2
€
0
TABLE XXIX. REDUCTIONS IN COUNTING- STRAYER-UPTON
The upper figures show descending reductions; the
lower figures descending ref^uctions. The frequency of
reductions ap'-'earing in examples is shown in the upper
left of each cell; those appearing in written problems
are shown in the lower right of each cell.
Gross
Score
Ream
Dozen
Quire
3
1
2
6
Unit
1
Sheet
Gross
Ream
Score
Quire
Dozen
Shee
Unit
1
5
i
42
TABLE XXX. REDUCTIONS IN TIME- STRAYER-UPTON
Century
Year
Month
The upner figure shows descending reductions;
the lower figure ascending reductions. The
frequency of reductions appearing in examples
is shown in the upper left of each cell; those
appearing in written problems are shown in the
lower risht of each cell. Further details
as to meaning and reading of table may be
found on page 32.
Week
Day
Hour
14
Minute
Second
Year
Month
Week
Day
Hour
^inute
8
10
Second
#
#
43
TABLE XXXI . REDUCTIONS IN MEASURES OF CONTENT AND CAPACITY'
3TRAYER-UPT0N
The upper figure shows reductions from pints, gallons,
etc. to cubic feet and cubic inches and the frequency
of their occurrence* The lower figure shows the reverse,
that is, the changes of cubic feet and cubic inches to
pints, gallons etc. Those occurring in examples are shown
in the upper left of the cell; those occurring in
written problems in the lower right of each ceil.
00
0)
d
I-H
0)
0)
o
0)
t3
CO
-p
rH
XI
C
C
C
i-H
0)
=J
o
cd
3
O
O
PQ
CU
S
1
1
Cubic Feet
Cubic Inches
2
00
CO
c
I-H
0)
o
0)
tJ
■p
I-H
-C
C
ra
c
r-i
OT
C
•H
as
O
o
O
PU
Cubic Feet
1
16
6
5
21
2
Cubic inches
1
2
44
TABLE XXXII. REDUCTION'S IN METRIC MEASURES- STRAYER-UPTON
Reductions from metric measures to Enp.lish equivalents
and visa versa. The fortner are shown in the upper fif^ures,
namely in fig. 1, fig. 5 and fig, 6; the latter in the
lower figures, fig. 4, fig. 7 and fig. 8.
Km. Dm. m, dm.
Miles
Yards
Feet
Inches
Fig. 1. Length
Fig. 2
Km. Dm. m. dm.
Miles
Yards
Feet
Inches
Fig. 3.
Fig. 4.
c
TABLE XXXII. (Continued)
Liter HI.
Quart
M.T. Kg,
Gallon
Pound
Bushel
Ton
Fig, 5. Capacity
Fig, 6. Weight
Liter HI.
M.T. Kg.
Pound
Ton
Quart
Gallon
Bu shel
Fig. 7, Weight
Fig, 8. Capacity
46
TABLE XXXIII. REDUCTIONS IN MISCEL- ANE0U3 MEA3URE3-
STRAYER-UPTON
Fig, 1 and 2 show chanp;es from knots, leagues to miles;
and bushels, pecks to pounds and ounces; fig. 3 and 4 show
the reverse process, namely changes from pounds, ounces to
bushels, pecks etc.
CO
(0
r-i
C
i-H
<D
w
0)
o
0)
CO
x:
-P
0)
0)
o
c
r-i
i-H
•r-l
(d
cd
PQ
(X.
CQ
Pounds
Ounces
m
o
0)
D
0)
ii-iiles
Fig. 1
Fig. 2
Pounds
(0
0)
0)
iH
C
l-l
0)
m
to
o
,— 1
U
CD
(0
o
c
l-l
u
rH
3
•H
cd
CO
CQ
{I.
OQ
CQ
Ounces
Fig. 3
Fig. 4
47
COMPUTATION WITH DENOMINATE NUMBERS
Generally speaking, denominate numbers are of two
kinds, namely, simple denominate numbers and compound
denominate numbers. vSimple denominate numbers consist
of units of measure of but one denomination, as, 1 inch.
Compound denominate numbers consist of two or more units
of measure of the same kind, such as, 2 yards 3 feet
5 inches. Lennes calls numbers involving two or more
units of different kinds, as for example those expressed
in "miles oer hour", "ton-miles" etc. "quasi-denominate"
1
numbers.
Computation in the fundamental processes involving
these three types of denominate numbers and the frequency
of their occurrence are shown in Tables XXXIV to XXXVI,
Written problems in connection with the clock and
the calendar which involve computation, are grouped
separately under Time,
1. Lennes, N» J. The Teaching of Arithmetic, page 281.
48
TABLE XXXIV. COMPUTATION WITH SIMPLE DENOMINATE NUMBERS
STRAYER- UPTON
The first column marked "Addition" shows the number of
occurrences of computation in adding the denominate numbers
involving the units of measure listed to the left of the
figure; the second column shows the number of occurrences
in subtraction, etc. Examples snown in upper left of cell;
written problems in lower right of each cell.
c
Inches
Feet
Yards
Miles
Acres
Square *"iles
Quart (Dry)
Peck
Bushel
Pint (Liquid)
Gallon
^arrel
C
o
o
u
CO
I
•H C
rH O
Q-, rH
•H -P
P
c
o
•r-l
•rH
>
Q
16
6
4
8
S
1
13
18
12
1
9
1
I
•5
2
11_
35
9
2
4
1
1
!
»
i
3
1 1
L
J
1
. 2
r —
1
i
1
I
f
)
TABLE XXXIV (Continued)
d
o
c
+>
o
(d
•fH
o
c
•P
•H
o
o
o
(d
cu
+>
'H
(D
•H
•p
•P
•H
n
iH
>
•i-t
<
03
3
P
Ounce
2
Pound
1
1
37
18
i
14
Dozen
1
Seconds
1
4.
3
ii'iinutes
2
3
3
2
Hours
i
...15
S
5
3
Days
L ■ -
i ^
3
Years
Clock-Calendar
iwinutes-"ec .
L
.
I 1
r
Hours-Min.
4
9
2
!
i
Month-Day
1
«
f
Yr.Mo.Day
il
i
c
TABLE XXXIV (Continued)
c
o
<
Degree (Temp)
Cu.Ft. (gas)
Kilowatt
c
O
1-1
c
■P
o
o
+>
C
o
r-4
o
L.
•r-l
m
+3
1-1
^
i-l
>
iH
(0
5
10
1
31
!
1
51
TABLE XXXV. COMPUTATION WITH "QUASI-DENOMINATE" NUMBERS-
STRAYER-UPTON
The first column marked "Addition" shows the number of
occurrences of computation in addinp^ the denominate numbers
involving the units of measure listed to the left of the
figure; the second column shows the number of subtraction
computations, etc. Examples are shown in upper left of the
cell; written problems in lov.er right of each cell.
Feet per second
Yards per second
Miles per second
Miles per minute
Miles per hour
Miles per day
Knots per hour
Km. per hour
Miles per gallon
Bushels per acre
c
i C
o
f-i
o o
c
■p
c
o
o
r-* -P
o
ft
«D
a.
+>
U
w
f-i
+>
x>
r-l
>
•d
3
3
<
03
P
1
1
2
10
1
1
1
6
i
1
2
18
36
4
3
14
1
1
1
1 — ^
L .,
i .
1
1
|_.._.
__J
j
!
21
1
L
Bosion University
t
TABLE XXXVI. COMPUTATION WITH COMPOUND DENOMINATE NUMBER3-
STRAYER-UPTON
The first column marked "Addition" shows the number of
occurrences of computation in adding the denominate numbers
involving the units of measure listed to the left of the
figure; the second column shows the number of subtraction
computations, etc. Examples are shown in upper left of the
cell; written problems in lower right of each cell.
Feet inches
Yards inches
Yards feet
Mi .rd. yd. ft. in.
Bushels pecks
Pecks quarts
Gallons quarts
Pounds ounces
Days hours ■
Hours minutes
C
O
c
4^
o
cd
o
c:
•p
t-i
c
O
o
o
t-i
cS
a.
•i-i
+^
u
•H
0)
•i-i
+>
+>
•r-l
•a
X>
r-l
>
3
3
<
to
a
Q
22
I. A...
! 1 i 1
I 2_
1
L...
■2
1 "iT
I
2 »
c
I
53
SUMMARY OF THE STRAYER-UPTON ARITHMETIC SERIES
In Table xxxvii we have a summary of the data gathered with
reference to the denominate numbers found in the Strayer-Upton
Arithmetic Series. The data has been gathered under five
headings. One group of facts has been omitted from this
summary, namely, "Informational Matter With Reference To
Denominate Numbers".. The reader may refer to pages 19 to 21
for this data.
Reading and Meaning of Table XXXVII.
The tables of measures involved in the summary are listed
to the left of the figure. Column 1 shows the number of
equalities found in each of the tables; column 2 shows the
number of units of measure used as mere words, while column 3
shows the number of written problems involved; in column 4
we find the number of table facts asked in examples and written
problems and column 5 shows the number of examples and written
problems involved; column 6 shows the total number of reductions,
while column 7 shows the number of examples and written
problems involved in these reductions; column 8 shows the
number of units of measure used in computation and column 9
the number of examples and written problems involved in
computation with simple denominate numbers; column 10 shows
the number of compound denominate numbers used and the final
column the number of examples and written problems involved
in computation with compound denominate numbers.
( (
54
SUMMARY OF STRAYER-UPTON. TABLE XXXVII.
For reading and meaning oi table, see page 53.
U. S. Money
Linear
Square
Cubic
Dry
Liquid
Weight
Counting
Time
Clock
Calendar
Capacity
Content
Metric
English
Equivalents
QuasiTDen.^No,
Misc*lileasures
m
0)
tH
rH
cu
a
B
U
(fi
cr
• o
X
(xj
rH
to.
0}
[
l5
c
T
•P
W
o
*rl
0)
cd
CO
o
CO
-P
m
c
(D
(D
•rH
<0
<D
rH
«-H
-P
rH
■p
i-H
o
a.
0)
a.
O
a.
cu
E
rH
e
C3
D.
B
s
«J
(tj
S
c6
o
X
00
X
X
o
X
o
W
w
w
o
w
o
rH
in
«>-
CO
rH
rH
6
6
1?
14
27
\
7
4
LOB
6
14
p
15
50
16
4
4
l45
4
15
30
6
2
17
2
2
11
14
4
2
6
2
1
1
4
8
3
2
8
> '
1
1
9
21
18
3
A
2
8
2
3
21
2
p
10
31
3
3
1
1
3
3
91
5
61
2
2
1
3
lo
7
2
61
5
4
1
7
10
7
184
3
16
43
5
vjVJ
2
4
7
4
20
10
8
59
10
3
5
10
20
2
6
3
5,Q
(
c
. J. . _
» _ . • . - . ... f .Ik-.-..,. » . . H.
1
c
55
TABLE XXXVIIA. DETAILED SUMMARY OF REDUCTIONS- Strayer-Upton
Column one shows the total number of descending reductions
with reference to each table of measure listed to the left
of the figure. The second column shows the total number of
examples (upper left of cell) and the total number of written
problems (lower right of cell). Column 3 and 4 show similar
data for ascending reductions.
+j •
o -P
3 o
xi 3
Q) to 'C5 0)
<D <1> 0)
r-J K r-l
m Ci. CL.
OS'S
CO flj o <d
(D X 0) X
Q cd << (a
United States "^oney
7
14
a
7
13
3
Linear Measure
9
30
6
6
20
10
Square Measure
6
8
6
6
4
Cubic Measure
2
4
2
4
Dry Measure
4
13
9
5
8
9
Liquid Measure
5
24
7
5
7
13
Weight Measure
> - - .
31
i_ ..
2
3
30
12
Counting
Time
.1..
10
25
_ .-8..
1
6
1
1
iT^
Engl»Eq»ofMetric Weights
4
1
9
' 1
(
--A
Capacity and Content
4
1
12.
6
7 !
i
I
Misc. Measures
7
6
4i
1
47 j
c
PART TWO
The following analysis covers the units of weight and
measure, with special reference to denominate numbers,
as found in the three book series of the New Everyday
Arithmetic, or as they will be called in this study
Hoyt-Peet.
i
< (
56
RELATIONS AMONG UNITS OF MEASURE
Tables XXXVIII - XLIX which follow show the various
tables of measures found in the New Everyday Arithmetic or
Hoyt-Peet, each Table representing; one table of measure,
except that counting articles and counting paper have been
grouped together in one table, as have also the various
measures of the Metric system. In each case the equalities
found in the table of measure has been stated to the left
of the figure and the frequency of occurrence of these
equalities is indicated in the upper left corner of the
corresponding square • All other relations found in the
texts of the Hoyt-Peet Series are listed to the right and
the frequency of occurrence of these equalities, either in
written problems or in some other isolated form, is
recorded in the lower right of the cell#
The Hoyt-Peet Series uses the puzzle element to
introduce tables of weight and measure* One of the
numbers in each relation of the table is omitted, for
instance, — inches — 1 foot",.^ The pupil is asked
to "complete the table".. This is merely a variation
in the method of introducing children to the relations in
the tables of weights and measures..
1.. Hoyt-Peet, Second Book, p. 144
i
< (
57
TABLE XXXVI II • RELATIONS AwlONG UNITS OF UNITED STATES MONEY
FOUND IN THE HOYT-PEET SERIES.
Equalities to the left are found in tables; frequency of
occurrence is shown in upper left of each cell. All other
equalities found in the texts are listed to the rip;ht;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
10 mills 7L 1 cent
5 cents st 1 nickel
10 cents = 1 dime
25 cents s 1 quarter
50 cents ^ § dollar
100 cents ss 1 dollar
10 dimes a 1 dollar
Total
1 mill ^ 1/10 cent
1 nickel = 5 cents
1 dime s 10 cents
1 quarter a 25 cents
5 dollar 50 cents
1 dollar = 100 cents
1 dollar s 10 dimes
58
TABLE XXXIX. RELATIONS Aiv"ONG UNITS OF LINEAR MEASURE
FOUND IN THE HCYT-PEET SERIES*
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence*
12 inches =. 1 foot
36 inches » 1 yard
3 feet =t 1 yard
16w feet - 1 rod
5280 feet s. 1 mile
5i yards s 1 rod
1760 yards s. 1 mils
320 rods = 1 mile
Total
1 foot - 12 inches
1 yard r 36 inches
1 yard ^ 3 feet
1 rod = 16|- feet
1 mile s 5280 feet
1 rod = 5|- yards
1 mile - 1760 yards
1 mile r 320 rods
59
TABLE XL. RELATIONS AwiONG UNITS IN SQUARE MEASURE
FOUND IN THE HOYT-PEET SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
144 sq»in,. = 1 sq»ft.
9 sq^ft* s 1 sq»yd*
30# sq^yd. s 1 sq.rd.
160 sq»rd,. =- 1 acre
43,560 sq»ft. s 1 acre
209 ft» square - 1 acre
640 acres = 1 sq»mi.
Total
1 3q»ft» =t 144 sq.in.
1 sq»yd. =t 9 sq»ft»
1 sq.rd* 30|- sq»yd,
1 acre s 160 sq»rd,
1 acre = 43,560 sq.ft,
1 acre - 809 ft. square
1 sq.mi. -5 640 acres
60
TABLE XLI.
RELATIONS AMONG UNITS IN CUBIC MEASURE
FOUND IN THE HOYT-PEST SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
I
1728 cu.in* z 1 cu.ft.
1 cu.ft. - 1728 cu.in.
1
27 cu.ft. =■ 1 cu.yd.
1 cu.yd. = 27 cu.ft.
1
128 cu.ft. = 1 cord
2
Total
1
61
TABLE XLII. RELATIONS AiTiONG UNITS IN DRY MEASURE
FOUND IN THE HOYT-PEET SERIES •
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
2 pints s 1 qt,
8 qt» = 1 pk,
4 pk. 3 1 bu.
Total
1 qt, - 2 pt,
1 pk. r 8 qt,
1 bu, z 4 pk.
#
62
TABLE XLIII. RELATIONS AMONG UNITS IN LIQUID MEASURE
FOUND IN THE HOYT-PEET SERIES •
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
2 pt* s 1 qt.
4 qt» » 1 gal.
Total
1 qt. = 2 pt.
1 gal. r 4 qt,
63
TABLE XLIV. RELATIONS AMONG UNITS IN MEASURES OF WEIGHT
FOUND IN THE HOYT-PEET SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the ri^^ht;
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
16 oz» s I lb.
100 lb» X 1 cwt»
2000 lb. = 1 T.
Total
1 lb» 16 02.
1 cwt. s. 100 lb.
1 T» -s 2000 lb.
64
TABLE XLV. RELATIONS AMONG UNITS IN TABLES OF COUNTING
FOUND IN THE HOYT-PEET SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell*
All other equalities found in the texts are listed to the right;
frequency of occurrences is shown in lower ri,a;ht of cell*
Blank indicates no occurrence*
12 units - 1 doz*
144 units - 1 gross
12 doz* - 1 gross
20 units - 1 score
Total
1 doz* - 12 units
1 gross - 144 units
1 gross - 12 doz*
1 score - 20 units
24 sheets - 1 quire
480 C500) sheets - 1 ream
20 quires - 1 ream
1 quire - 24 sheets
1 ream - 480 (500) sheets
1 ream - 20 quires
t
■J
65
TABLE XL VI,. RELATIONS AMONG UNITS IN MEASURES OF TIME
FOUND IN THE HOYT-PEST SERIES.
Equalities to the left are found in tables of measures;
frequency of occurrence is shown in upper left of each cell.
All other equalities found in the texts are listed to the right
frequency of occurrence is shown in lower right of cell.
Blank indicates no occurrence.
60 seconds - 1 minute
60 minutes - 1 hour
24 hours - 1 day
7 days - 1 week
365 days - 1 year
366 days - 1 leap year
52 weeks - 1 year
12 months - 1 year
100 years - 1 century
Total
1 minute - 60 seconds
1 hour - 60 minutes
1 day - 24 hours
1 week - 7 days
1 year - 365 days
1 leap year - 366 days
4 weeks - 1 month
1 year - 52 weeks
1 year - 12 months
1 century - 100 years
(#
66
TABLE XL VII. RELATIONS AMONG UNITS OF METRIC MEASURES
FOUND IN THE HOYT-PEET SERIES.
Equalities to the left are found in tables of measures.
Frequency of occurrence of these table facts is noted in
upper left of cell.
10 mm. s 1 cm*
10 cm» :t 1 dm,
10 dm* s. 1 m*
10 m, =1 Dm.
10 Dm. - 1 Hm.
10 Hm. - 1 Km.
10 Km. ^ 1 Mm.
Total
1
1
1
1
1
7
Fig* 1. Length
(•
TABLE XL VI I (CONTINUATION)
Equalities to the left are found in tables of measures
Frequency of occurrence of these table facts is noted
upper left of cell.
100 sq» mm,. = 1 sq» cm,
100 sq, cm, - 1 sq» dm,
100 sq, dm, ^1 sq, m,
100 sq, m. r 1 sq. Dm,
100 sq. Dm, = 1 sq, Hm,
100 sq, Hm. - 1 Km,
Total
Fig, 2, Square Measure
TABLE XL VI I (CONTINUATION)
Equalities to the left are found in tables of measures
Frequency of occurrence of these table facts is noted
upper left of cell.
L
1000 cu. rnrn* -t 1 cu, cm.
1
1000 cu. cm» » 1 cu.. dm.
1
1000 cu. dm. - 1 cu. m.
3
Total
Fig. 3. Cubic Measure
TABLE XL VI I (CONTINUATION)
Equalities to the left are found in tables of measures
Frequency of occurrence of these table facts is noted
upper left of cell.
10 ml, 51 1 cl.
10 cl, = 1 dl.
10 dl. =11.
10 1. = 1 Dl.
10 Dl. = 1 HI.
Total
1
1
Fig. 4. Capacity
TABLE XL VI I (CONTINUATION)
Equalities to the left are found in tables of measures
Frequency of occurrence of these table facts is noted
upper left of cell.
10 cg» s 1 dg,.
10 dg» rig.
10 g» = 1 Dg.
10 Dg. s 1 Hg.
10 Hg» t;. 1 Kg,
10 Kg. - 1 Mg..
10 Mg. r 1
10 Q.. r 1 M.T.
Total
1
1
1
1
1
1
1
8
Fig. 5. Weight
1
71
TABLE XL VII I • RELATIONS BETWEEN UNITS OF METRIC MEASURES
AND ENGLISH MEASURES FOUND IN HOYT-PEET.
Equalities to the left are found in table of measure.
Frequency of occurrence of these table facts is noted in
upper left of cell.
1 m» •= 39»37 in,
1 Km, - ,62 mi, 5/8 mi,
1 Ha, - 2,47 acres
1 1, :r ,91 qt, 9Dry)
1 1, 5t 1.06 qt, (Liquid)
1 HI, =. 2,84 bu,
1 Kg, = 2,2 lb,
1 M,T, - 1.1 T
Total
8
72
TABLE XLIX^ RELATIONS IN MISCELLANEOUS MEASURES
FOUND IN THE HOYT-PEET SERIES •
This table shows relations in miscellaneous measures
and the frequency of their occurrence in the Hoyt-Peet
Ari thmetics*
Linear Measure
1 Knot = 1»1516 statute miles 1
Capacity and Content
1 cu» ft» water = 62|- lb» 1
12 cu» ft» air = 1 lb. 1
5 cu, ft, wheat s 4 bu. 1
5 cu. ft, ear corn s 2 bu, 1
1 bu. = 2150.42 cu, in. 3
1 bu. jz 1 i cu. ft. I
1 bu. corn (shelled) ^ 56 lb. 8
I bu. potatoes ^ 60 lb. L
1 gal. water s 231 cu, in. 3
1 bbl. flour r: 196 lb. 1
1 glass s k pt* 1
Weight
1 long ton :t 2240 lb. 1
1 short ton = 2000 lb. 1
360 degrees = 1 circle
2
73
INFORMATIONAL MATTER WITH REFERENCE TO UNITS OF MEASURES.
Since the actual use of units of weight and measure is
always in connection with articles and commodities, an under-
standing of the meaning and use of units of weight and measure
can only come through commodities* As soon as a commodity
is thoroughly understood, the appropriate unit of measure to
be used in connection with it has been learned incidentally,.
Units of weight and measure are not learned as such, but in
connection with commodities* It is this idea of the con-
nection or inter-relation between commodities and units of
weight and measure that the expression "informational matter",
used here in connection with denominate numbers, is meant to
convey. No manipulation or computation with denominate
numbers is involved > but simply an understanding that a given
unit of measure is used in connection with a given commodity.
The instances in which nothing more is involved than an
understanding of the meaning of a unit of measure through its
use with a commodity, have been gathered and grouped under
two headings, namely, those appearing in statement form and
those appearing in question form*
74
For instance, the statement is made that "In measuring land
and other surfaces, the square inch, the square foot, and the
square yard are used^"'^ The information is imparted that square
measures are used to measure surfaces and is, therefore, listed
under "Informational Matter in Statement Form**,. Again, the
question is asked, "What are the measures that are commonly
used for apples, potatoes, wheat, and other fruits, vegetables,
and grains?"^ This is placed under the heading of "Informa-
tional Matter in Question Form,
Informational Matter in Statement Form found in
Hoyt-Pee t.
Money:
The mill, the thousandths part of a dollar, is
often used in the calculations of business men
and public officials^
The standard foreign coins are the following:
British pound sterling
British shilling
German mark
French franc
Italian lira
Russian ruble
Spanish peseta
Mexican peso
Square Measure:
In measuring surfaces the square units used are
the square inch, the square foot, the square yard,
the square rod, the acre, and the square mile»
1. Hoyt-Peet, Second Book, page 148.
2^ H It II •• n 3^51^
75
Cubic Measure:
In measuring the contents of a box, a grain bin,
a wagon, or a freight car, cubic units are some-
times used*
Weight Measurer
Freight is shipped by the hundredweight; also by
the net ton (20001b») and the gross ton (2240)lb,
Liquid Measure:
Milk is bought in pint or quart bottles.
Counting:
Shoes are bought in pairs*
Time r
A month counting 30 or 31 days (28 or 29 in Feb.)
is called a calendar month; one counting 4 weeks is
called a lunar month.
Miscellaneous Measures:
Angles are measured by degrees;
Electrical energy is measured by the watt or kilowatt.
Origin of the thermometer and the reason for
dividing it as it is.
Metric measures:
In measuring land, the square Dm. is called an
are; the square Hm. is called hektare.
In measuring wood the cubic meter is called a stere*
How the length of the meter was arrived at.
Informational Matter in Question Form found in Hoyt-Peet.
Name the coins used in buying and selling;
What unit of measure do you use in measuring height?
What measure is used in selling cloth?
Name measure used in finding the length of cloth;
the width?
What measure is used for the depth of a building lot?
What measure is used for the width of a field?
What measure is used for finding distances between
cities?
c
76
What unit is used for railway distances?
Pace and measure a distance one rod long,.
Name a place 1 mile from school.
Which units would one use in measuring the surface of
a small sheet of silver?
In measuring the area of a room; of a sidewalk; of a
field, of a building lot, of a farm? Which
unit is used for the area of a continent or
an ocean?
What unit of measure is used for apples, potatoes,
wheat, oats, fruits, vegetables, and grains?
What unit of measure is used in computing quantity of
earth excavated?
By what weight is butter sold?
What unit of measure is used in measuring one's
weight?
What units of me-^sures are used in mixing a cake?
in canning fruit?
What unit of weight is used for first call mail;
for parcel post; for hay?
By what unit of measure are eggs sold? name fruits
that are sold by the same measure.
In selling screws and other small articles, what
name is given to 12 dozen?
What units of measure are used for giving one's age?
Name units of measure used for each of the following
electricity, area of a farm, area of a room,
coal, wheat, potatoes, oranges, sugar, hay,
strawberries, peaches, time, gas, rainfall,
temperature, land, distance at sea, speed of
a ship, height above sea level?
77
UNITS OF MEASURE MERELY AS NAMES
Many of the written problems contain units of
measure which, however, involve no computation or
manipulation with denominate numbers or such.
The units of measure are mentioned merely as words.
The following problems might be cited to show the type
of written problem involved: "Some of the fast vessels
in the United States Navy travel at the rate of 22.5
1
knots or 25.911 miles, per hour." "Two pounds of
candy are to be put into boxes holding half pound each.
2
How many boxes are needed?" "A railway was built
at the cost of |24,000 per mile. At that rate,
how many miles of railway can be built for $2,400,000?"
1. Hovt-^Peet. Second Book, page 203, §\&
2. Ibid. page 135, ^\
3. " "83. #18
0-
78
TABLE L. UNITS OF MEASURE A3 WORDS IN HOYT-PEET
The table shows the frequency of occurrence of units
of measure involving no computation with denominate numbers
as such.
Linear Measure
Inches
Feet
Yards
Miles
Square Measure
Acres
Sq, iJ'iiles
Cubic Measure
Cord
Dry ii^easure
Quarts
Pecks
Bushels
Liquid Measure
Pints
Quarts
Gallons
38
29
112
35
17
5
3
2
34
3
16
12
Weip:ht "^easure
Pounds
Ounces
Ton
Counting
Pair
Dozen
Time
121
1
13
35
62
Seconds 1
"^inutes 40
Hours 61
Days 44
n'eek 50
""onth 21
Year 24
Miscellaneous Measures
Knots 1
Cu, ft. (gas) 3
Kwt. ' 8
Barrel 17
Bales 3
TABLE FACTS IN EXAMPLES AND WRITTEN PROBLEMS
Tables LI to LIX show the number of times a table
fact or the reverse statement is called for either in
examples or in written problems. For example, a table
fact may be called for in an example of the following
type, "How many inches in one foot?" Or a written
problem may involve the knowledsze of a table fact so
that it may be solved, as for instance in the following
written problem: "Mr. Archer finds that the distance
around the deck of a steamsnip os 264 ft. How many
times must he ro around the deck to walk a mile?
To solve this problem involves a knowledge of the fact
that there are 5280 feet in one mile.
80
TABLE LI. TABLE FACTS IN UNITED STATES MONEY ASKED IN
EXAMPLES AND WRITTEN PROBLEMS - HOYT-PEET
In each instance, the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 10 mills equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of figure thus: What is one nickel equal to?
Frequency of occurrence is shown in the column of cells
to the right. The frequency of occurrence in examples
is shown in the upper left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
10 mills =(1 cent)
5 cents =. (1 nickel)
10 cents 5 (1 dime)
25 cents s (1 quarter)
50 cents r (I dollar)
100 cents *(1 dollar)
10 dimes =.(1 dollar)
13
I
18
1
21
PI
7
1 mill r (1/10 cent)
1 nickel =. (5 cents)
1 dime ^ (10 cents)
1 quarter - (25 cents)
I dollar = (50 cents)
1 dollar (100 cents)
1 dollar - (10 dimes)
81
TABLE LI I. TABLE FACTS IN LINEAR MEASURE ASKED IN
EXAMPLES AND WRITTEN PROBLEMS- HOYT-PEET
In each inst-^ince the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus :
What are 12 inches equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
tight of the figure thus: What is one foot equal to or
how many inches are there in one foot? Frequency of
occurrence is shown in the column of cells to the right.
The frequency of occurrence in examples is shown in the
upper left of the cell; the frequency of occurrence in
written problems is shown in the lower right of cell, ^lank
indicates no occurrence.
12 inches* (l foot)
3 feetr (l yard)
16^ Feetr (l rod)
5280 feetr (1 mile)
5| yards- (1 rod)
1760 yards-(l mile)
320 rodsT. (1 mile)
1 foot:: (12 inches)
1 yardz (3 feet)
1 rod^ (16| feet)
1 miler (5280 feet)
1 rod's (5| yards)
1 miles (1760 yards)
1 mile-s (320 rods)
t
82
TABLE LIII. TABLE FACTS IN SQUARE MEASURE ASKED IN
EXAMPLES AND WRITTEN PROBLEMS- HOYT-PEET
In each instance the fact in parenttjesis indicates
the fact asked for. Read to the left of the figure thus:
What are 144 square inches equal to? Frequency of occurrence
is shown in the column of cells to the left. Read to the
right of the figure thus: What is 1 square foot equal to?
Frequency of occurrence is shown in the column of cells to
the right. The frequency of occurrence in examples is
shown in the upper left of the cell; the frequency of
occurrence in written problems is shown in the lower right
of each cell.
144 sq. in.=(l sq.ft.)
9 sq. ft. r (1 sq. yd.)
ZO^ sq. yd. s (1 sq. rd.)
160 sq. yd. -r (1 A. )
640 A. = (1 sq. mi . )
36 sq. mi. - (1 tp.)
1 sq. ft. ^(144 sq. in.)
1 sq. yd, c(9 sq. ft.)
1 sq. rd. - (30i sq. yd.)
1 A. i (160 sq. yd. )
1 sq. mi. - (640 A. )
1 tp. r (36 sq. mi . )
83
TABLE LIV, TABLE FACTS IN CUBIC MEASURE ASKED IN
EXAMPLES AND WRITTEN PROBLEMS - HOYT-PEET
In each instance, the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 1728 cu. in. equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
ri)!7ht of the fif?ure thus: How many cu. in. in one cu. ft.?
Frequency of occurrence is shown in the column of cells
to the rif?ht. The frequency of occurrence in examples
is shown in the upper left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
1728 cu. in = ( 1 cu.f t. )
27 cu. ft. -z. (1 cu. yd. )
1
1
1 cu. ft. T (1728 cu. in.)
1 cu. yd. r (27 cu. ft. )
84
TABLE LV. TABLE FACTS IN DRY MEASURE ASKED IN EXAI»/IPLES
AND WRITTEN PROBLEMS - HOYT-PEET
In each instance the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 2 pints equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
ri^^ht of the figure thus: How many pints are there in
one quart? Frequency of occurrence is shown in the
column of cells to the right. The frequency of occurrence
in examples is shown in the upper left of the cell; the
frequency of occurrence in written problems is shown
in the lower rio'ht of esch cell. Blank indicates no
occurrence .
2 pins =. (1 quart)
8 quarts = (1 peck)
4 Decks =. (1 bushel)
2
5
5
4
n
1 quart ^ (2 pints)
1 peck r (8 quarts)
1 bushel = (4 pecks)
85
TABLE LVI. TABLE FACTS IN LIQUID MEASURE ASKED IN EXAMPLES
AND \VRITTEN PROBLEMS - HOYT-PEET
In each instance the fact in parenthesis indicates the
fact asked for. Read to the left of the fiG;ure thus:
What are 2 pints equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: One quart is equal to how many
pints? Frequency of occurrence is shown in the column of
cells to the right. The frequency of occurrence in
examples is shown in the upper left of the cell;) the
frequency of occurrence in written problems is shown
in the lower ris^ht of each cell; Blank indicates no
occurrence ,
2 pints :5 (1 quart)
4 quarts - (l gallon)
1 quart = (2 pints)
1 gallon c (4 quarts)
0^
86
TABLE LVII. TABLE FACTS IN MEASURE OF WEIGHT ASKED IN EXAIvlPLES
AND WRITTEN PROBLEMS - HOYT-PEST
In each instance, the fact in oarenthesis indicates
the fact asked for. Read to the left of the fip;ure thus:
What are 16 oz. equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: How many oz. in 1 pound?
Frequency of occurrence is shown in the column of cells
to the right. The frequency of occurrence in examples
is shown in the upper left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
16 ounces s (1 pound)
100 lb. s(l cwt. )
2000 pounds = (1 ton)
1 pound - (16 ounces)
1 cwt. (100 lb.)
2000 pounds = (l ton)
87
TABLE LVIII. TABLE r-'^ACTS IN COUNTING ASKED IN EXAMPLES
AND WRITTEN PROBLEMS - HOYT-PEET
In each instance, the fact in parenthesis indicates
the fact asked for. Read to the left of the figure thus:
What are 12 equal to? Frequency of occurrence is
shown in the column of cells to the left. Read to the
right of the figure thus: How many equal one dozen?
Frequency of occurrence is shown in the column of cells
to the right. The frequency of occurrence in examples
is shown in the upper left of the cell; the frequency
of occurrence in written problems is shown in the lower
right of each cell. Blank indicates no occurrence.
3
12 = (1 dozen)
16
144 :r (1 gross)
12 dozen r (1 <2ross)
1 dozen - (12)
1 gross - (144)
1 gross = (12 dozen)
88
TABLE LIX. TABLE FACTS IN TIMS ASKED IN EXAMPLES AND
WRITTEN PROBLEMS - HOYT-PEET
In each instance the fact in parenthesis indicates the
fact asked for. Read to tne left of the figure thus: What
are 60 seconds equal to? Frequency of occurrence is shown
in the colu;nn of cells to the left. Read to the ri/ht of
the fi'^ure thus: How many seconds in 1 minute? Frequency
of occurrence is shown in the column of cells to the right.
The frequency of occurrence in examples is shown in the
upper left of the cell; the frequency of occurrence in
written problems is shown in the lower right of each cell.
Blank indicates no occurrence.
60 seconds - (1 minute)
60 minutes =. (l hour)
24 hours = (1 day)
7 days - (1 week)
365 days - (1 year)
366 days -(l Ip. year)
52 weeks - (1 year)
12 months - (I year)
100 years - (1 century)
1
1
3
5
1
1
2
3
i ■
11
1 minute (60 seconds)
1 hour r (60 minutes)
1 day - (24 hours)
1 week z (7 days)
1 year z (365 days)
1 Ip. year =. (366 days)
1 year - (52 weeks)
1 year - (12 months)
1 century - (100 years)
0*
89
REDUCTION OF DENOMINATE NUMBERS
Reduction of denominate numbers is the process of
changing their denomination without changing their value.
Denominate numbers may be changed from a higher to a
lower denomination, as for example, when Z miles are
reduced to feet; denominate numbers may also be
changed from a lower to a higher denomination, as when
21 pints are reduced to gallons. The former process is
known as descending reduction; the latter, as ascending
reduction.
Meaning and Reading of Tables LXII to LXXIII.
The upper figure shows descending reductions. The
unit of measure at the head of the column of vertical
squares in these fi.^ures read in connection with each
of the cells in these columns indicate the denomination
from which the reduction is made; when these terms are
read in connection with the row of horizontal cells to
the left, they indicate the denomination to which the
reduction is made.- The lower figures show ascending
reductions. The unit of measure read in connection with
the horizontal rov.s of cells to the right indicate the
denomination from which the reduction is made; read in
90
connection with the vertical column of cells it refers
to the unit to which the ascending reduction is made.
Frequency of occurrence of reductions appearing in
examples is shown in the upper left of each cell; those
appearing? in written problems are shown in the lower
risht of each cell.
91
TABLE LX. REDUCTIONS IN UNITED STATES MONEY-HOYT-PEET
The upper figure shows descending reductions; the lower
figure ascending reductions. The frequency of reductions
appearing in examples is shewn in the upper left of cell;
those appearing in written problems in lower right. For
further details as to meaning and reading of table see page 89.
Dollars
Half dollars
Quarters
Dimes
28
11
Nickels
21
Cents
Mills
Dollars
Half dollars
Quarters
Dimes
Nickels
Cents
5
5
2
Mills
92
TABLE LXI. REDUCTIONS IN LINEAR MEASURE - HOYT-PEET
The upper figure shows descending reductions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each square; those appearing in written
problems are snown in the lower right of each square.
For further details as to meaning and reading of tables
see page 89,
Miles
TT
Rods
Yards
46
49
Feet
43
Inches
Vli les
Rods
1
Yards
1
Feet
24 i 4
!
L 311
8
Inches
20
10
19
1
93
TABLE LXII. REDUCTIONS IN SQUARE MEASURE - HOYT-PEET
The upper fiejure shows descending redactions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each square; those appearing in written oroblems
are shown in the lower right of each square. For further
details as to meanin.? and reading of table see page 89.
Square Miles
Acres
5
2
Square Rods
1
1
Square Yards
. I,...
2
3
1
6
2
Square Feet
I
Square
Square Miles
Square
Acres
1
Square
Rods
2
6
Square Yards
1
1
Square Feet
4
7
1
1
1
5
Inches
2
TABLE LXIII. REDUCTION?? IN CUBIC MEASURE - HOYT-PEET
The upper fip;ure shows descending reductions; the
lower figure ascendinp reductions. The frequency of
reductions appearing? in examples is shown in the
upper left of each cell; those appearin,-? in written
problems are shown in the lovi/er right of each cell.
Cords
Cubic Yards
4
Cubic Feet
2
1-
Cubic Inches
Cords
Cubic Yards
Cubic
Feet
5
8
1
Cubic Inches
2
1
95
TABLE LXIV. REDUCTIONS IN DRY MEASURE- HOYT-PEET
The upper fifj-ure shows def^cendinp; redactions; the
lower figure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each square; those appearing in written problems
are shown in the lower right of each square. For further
details as to reading and meaning of table see page 89,
Bushels
s
2
Pecks
1
2
Quarts
Pints
Bushels
Pecks
3 '
Quarts
1
2
1
■
i
i
Pints
€
8
96
TABLE LXV. REDUCTIONS IN LIQUID MEASURE - HOYT-PEET
The upper figure shows descending redactions; the
lower fi/^ure ascending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each cell; those annearin-^ in written oroblems
are shown in the lower right of each cell. For further
details as to reading and meanin.? of table see page 89.
Barrels
■' (rallons
Quarts
Pints
Cups or Glasses
Barrels
Gallons
Quarts
Pints
97
TABLE LXVI. REDUCTIONS IN WEIGHT MEASURES - HOYT-PSET
The UDDer fierure show?? descendin«y reductions; the
lower figure ascendirif^ redactions. The frequency of
reductions appearins- in examples is shown in the upper
left of each square; those appearin-? in written problems
are shown in the lower right of each square. For further
details as to reading and meaning'- of table see page 89,
T:ons
8
Cwt .
Pounds
15
Ounces
Tons
r
Cwt.
Pounds
3
3
Ounces
10
6
98
TABLE LXVII. REDUCTIONS IN COUNTINO - HOYT-PEET
The upner figures show descendine reductions; the
lower figures descending reductions. The frequency of
reductions appearing in examples is shown in the upper
left of each cell; those appearin.-'- in written problems
are shown in the lower rip;ht of each cell.
Gross
Score
Ream
Dozen
1
13
17
Unit
Quire
Sheet
Gross
Ream
Score
-r
Quire
Dozen
•
Sheet
Unit
1
5
99
TABLE LXVIII. REDUCTIONS IN TIMS - HOYT-PEST
Century
Year
lonth
The upper figure shows descending reductions
the lower fip;ure ascendinp; reductions. The
frequency of reductions appearing in
examples is shown in the upper left
of each cell; those appearing in
written problems are shown in the
lower right of each cell.
Further details as to meaning
and reading of table may be
found on page 89.
Week
Day
Hour
13
Minute
Second
Century
Year
Month
Week
Day
Hour
Minute
Second
100
TABLE LXIX. REDUCTIONS IN MSASUR'"':S OF CONTENT AND CAPACITY-
HOYT-PEET
The upper figure shovvs reductions from pints and gallons
etc. to cubic feet and cubic inches and the frequency
of their occurrence. The lower figure snows the reverse,
that is, the changes of cubic feet and cubic inches to
pints, gallons, etc. Those occurring in examples are shown
in the upper left of the cell; those occurring in
written problems in the lower right of each cell.
Gallons
Bushels
Pounds
Tons
1
Cubic feet
Cubic inches
m
C
r^
m
o
a>
•d
Xi
C
m
iH
m
c
CS
r±
o
O
a.
Eh
Cubic Feet
Cubic Inches
101
TABLE LXX REDUCTIONS I'^ METRIC MEASURES- Hoyt-Peet
Reductions from metric measures to English equivalents
and visa versa. The former are shown in fig, 1,2,5 and 6;
the latter in fig. 3,4,7 and 8. For further details as to
the reading of the table refer to page 89,
Km. Dm. m. dm.
2
Miles
Y
ards
Feet
Inches
H
a. sQ.m.
Sq, mi.
Acres
Sq. Rods
Fig. 1, Length
Fig. 2. Square ^"^easure
Km. . Dm. m. dm.
m Ha,
Sq. Mi.
Acres
Sq. Rd.
whiles
ards
Feet
Inches
Fig. 3
fig. 4
TABLE LXX (Continued)
Liter HI.
1
1 1
Quart
M. t.
Kg.
1
Gal.
1
1
Pound
Bushels
1
Ton
Fig. 5 Fig. 6
Mt. Kg.
Pound
Ton
Fig. 7
Liter HI.
Quart
Gallon
Bushel
Fig. 3
c
103
TABLE LXXI REDUCTIONS IN FOREIGN MONEY- HOYT-PEET
The upper figure shows the number of examples and problems
involving changes from United States money to foreign money;
the lower figure the number involving changes from foreign
money to its equivalent in U.S. money.
United States Money
to
Pounds (English)
Francs
1 Marks
Lira
U.S. Money
Pounds to
Francs to
Marks to
t
104
COMPUTATION '.VITH DENOMINATE NUMBERS
Gener ally speaking, denominate numbers are of t'Ao
kinds, namely, simple denominate numbers and compound
denominate numbers. Simple denominate numbers consist
of units of measure of but one denomination, as, 1 inch.
Compound denominate numbers consist of two or more units
of measure of the same kind, such as, 8 yards 3 feet
5 inches. Lennes calls numbers involving two or more
units of different kinds, as for example those expressed
in •♦miles per hour", or "ton-miles" "quasi-denominate"
numbers.^
Computation in the fundamental processes involving these
three types of denominate numbers and the frequency of their
occurrence are snown in Tables LXXII to LXXIV.
L. Lennes, N.J. The Teaching of Arithmetic, p. 281
( (
i t
105
TABLE LXXII. COMPUTATION WITH SIMPLE DENOMINATE NUMBERS-
HOYT-PEET
Unit of measure involved noted on the left; the computation
involved noted at head of each column; frequency of occurrence
of examoles of each noted in upner left; written problem lower right.
Inches
Feet
Yards
Rods
Miles
Square Feet
Square Rods
Acres
Square Miles
Cubic Inches
Cubic Feet
Bushels
c
o
•H
-P
tH
•d
<
c
o
ft
+>
o
«J
u
CO
I
o C
f-t o
rH -H
CU-P
•rH
+J
rH
c
O
•H
CO
■r-i
>
Q
9
55
7
27
5
21
21
9
9
22
6
16
4
12
4
7
5
43
3
40
3
24
5
3
1
2
L -23.
1
23
i
1
1
3
1
1"!
»- . . .
2
9
5
„,.,
>
1
1
L.. ^
1
1
i
TABLE LXXII (Continued)
106
Pints
Gallons
Barrels
Ounces
Pounds
Hundredweight
Ton
Dozen
Seconds
Minutes
Hours
Days
Degrees
Cu. ft. (gas)
c
o
<
c
o
c
-p
Q
•H
o
-P
•H
C
o
e-H
0
<d
O4
•H
u
•^^
OT
-p
-p
JO
1-1
>
3
CO
Q
1
1
1
1
2
1
1
1
1
1
1
3
L -14..
3
8
4
16
2
1
jl
1
1
•
. A
2
... 5
2
1
1
i
1 1
' 1
i
1
.V
■ 1
1 1
, J
1 !
1
4
1
1
1
3
>
1
2
1
1
1
5
6*-
4
107
TABLE LXXII (Continued)
Same as previous, but units of time in connection with
the clock or the calendar.
c
*
o
ft
ta •
d
•p
c a.
o
o
•rH 1
•rH
cri
> •
-P
U
•H
+J
o •
n
X)
> <<
d
<
CO
n
Hour
7
2
nour "iinut.es
1
13
23
19
Month day
1
Yr.Mo. Day
8
Years
4
Mo. Day Hr. "^in.
1
>
1
i
1
1
/
▲
108
TABLE LXXIIt.
COMPUTATION WITH "QUASI-DENOMINATE" NUMBERS-
HOYT-PEET
The first column marked "Addition" shows the number of
occurrences of computation in adding?; the denominate numbers
involving the units of measure listed to the left of the figure;
the second column shows the number of subtraction computations , etc.
Examples are shown in upper left of the cell; written problems in
lower right.
c
o
•fH
-P
•iH
n
<
c
o
+>
o
(0
u
CO
c
o
■p
CO
o
D.
ft
+^
<-l
3
c
o
to
>
Q
Miles per minute
Miles per hour
Miles per day
Miles per day,hr,min
Nautical mi. per day
Miles per gallon
Bushels per acre
Tons oer acre
2
2
6
9
11
19
45
2
16
1
t
1
1
1
t 1
1
f i
i . . .i
3
10
2
5
109
TABLE LXXIY. COMPUTATION WITH COMPOUND DENOMINATE NUMBERS-
HOYT-PEET
The first column marked "Addition" shows the number of
occurrences of computation in addin^j; the denominate numbers
involving the units of measure listed to the left of the
figure; the second column shows the humber of subtraction
computations, etc. Examples are shown in upper left of each
cell; written problems in lower right.
c
o
c
■p
o
ClJ
o
-P
•H
C
o
o
O
•H
ex.
•H
+>
u
(0
l-l
>
3
•i-l
<
CO
Q
Feet inches
3
6
3
8
6
Yard inches
4
6
5
3
Yard feet
1
Rods feet
1
2
Miles Rods
1
2
Hours ^inutes
1
Sq, yd* Sq. ft.
1
Sq. ft. Sq.in.
1
Bushels pecks
1
110
SUMMARY OF THE HOYT-PEST ARITHMETIC SERIES
In Table LXXV we have a summary of the data gathered with
reference to the denominate numbers found in the Hoyt-f' eet
arithmetics. The data has been grouped under five headings.
One group of facts has been omitted from this summary, namely,
"Informational Matter with Reference to Denominate Numbers".
The reader may refer to pages 73-76 for this data.
Reading and Meaning of Table LXX.V: •
The tables of measures involved in the suminary are listed
to the left of the figure. Column 1 shows the number of
equalities found in each of the tables; column 2 shows the
number of units of measure used as mere wo^ds, while column 3
shows the number of written problems involved; in column 4
we find the number of table facts asked in examples and written
problems and column 5 shows the number of examples and
WEitten problems involved; column 6 shows the total number
of reductions, while column 7 shows the number of examples
and written problems involved in these reductions; column 8
shows the number of units of measure used in computation and
column 9 the number of examples and written problems involved
in computation with simple denominate numbers; column 10 shows
the number of compound denominate numbers used and the final
column shows the number of examplr=s and written problems involved
in computation with compound denominate numbers.
Ill
TABLE LXXV. SUMMARY OF HOYT-PEET ARITHMETICS
FOR reading and meaning of table see page 110,
U.S. Money
Linear Measure
Square Pleasure
Cubic Measure
Dry Measure
Liquid ^^easure
Weight Measure
Counting
Time
Clock
Calendar
Metric Measures
English
Equi valents
Foreign Exchange
Quasi-Den.No.
Misc. Measures
0)
CO
■p
01
o
f\
%j
■iH
CO
03
O
+>
(D
Ul,
<U
•H
1-1
rH
•H
-P
rH
C.
0)
Qu
O
a
T>
s
iH
E
3
U
03
X3
ClJ
O
X
CC
X
0)
W
(r:
rH
to
IT)
(0
0)
rH
Du
E
cc
X
w
c
o
iH
■p
■P
3
CL
E
O
P
(0
Q)
iH
&.
E
CO
X
c
3
o
E
o
o
CO cr>
10
(D
iH
a.
E
<d
X
W
4
59
IC
10
91
4
4
214
5
16
2
16
231
7"^
5
35
32S
5
48
3
2
22
14
18
29
4
3
16
2
2
2
1
1
2
2
5
20
11
2
2
3
3
39
r
3
11
16
9
16
4
1
1
1
1
1
2
3
31
2
6
1
5
18
13
3
4
4
3
3
135
3
6
8
40
23
4
17
52
3
2
97
2
3
17
5
18
29
1
1
6
7
246
7
9
20
7
26
11
4
7
13
6
37
85
29
8
17
21
6
7
13
5
8
2
134
15
5
32
2
11
6
s
112
TABLE LXXVa. DETAILED SUMMARY OF REDUCTION- HOYT -Peet,
Column one shows the total number of descending reductions
with reference to each table of measure listed to the left
of the figure. The second column shows the total number of
ascending reductions. Column 3 and 4 show similar data for
a'^.cending reductions.
United States ^oney
Linear Measure
Square Measure
Cubic Measure
Dry Measure
Liquid iVieasure
Weight Measure
Counting
Time
Engl. Eq. of Metr We.
Foreign Exchange
•
•
o
o
3
TJ
to
t3
CO
0)
0)
0)
<D
l-l
r-t
•
a.
a.
o
E
•
E
(0
o
c6
0)
x:
X
o
W
79
12
X 'J
3
?3
— H
7
159
7
72
23
50
16
12
7
7
8
21
■
6
14
2
3
1
10
10
6
3
•2,
t
3
1
1
9
9
i 3
2
1
i
9
4
1 ]
PA
16
4
♦
. . .
12
13
5
1 2
3
18
11
1 ,
20
! 4
3
10
1
^3
8
» 9
8
i
5
I
4
9
3
i ^
1
4
1
—
\
(5
nal Summary and Concluding Observations
113
TABLE LXXVI. FINAL SUMMARY
Tables ^
Number of tables in texts 16 13
Number of relations in tables 68 60
Number of Miscellaneous relations 20 15
Total number of relations taught 88 75
Units of measure merely mentioned 26 30
Table facts in e>:am|;^les and written problems
Number of table facts 20 28
Number of examples and problems involved 61 168
Reductions
Number of reductions 109 109
Examples and problems involved 482 639
Comoutation
Simple units of mea=;ure 23 27
Examples and problems involved 399 673
"Quasi-Denominate" numbers 10 8
Examples and problems involved 136 136
Compound denominate numbers 10 8
Examples and problems involved 73 54
114
Concluding Observations.
1. One notes the absence of many of the tables of
measurement found in older arithmetic text-books
but now generally considered obsolete. However,
the metric system is still retained in both series,
a doubtful procedure. The Upton-Strayer series
also retains the tables of so-called "useful
equivalents in liquid measure," called in this
study "measure of capacity," as well as the table
of "useful equivalents in dry measure," called in
this study, "measure of content."
2. Each series insists on the memorization of the entire
table of measure as such. The Fourth Yearbook of the
Department of Superintsndence stated that "tables
1
as such are unnecessary." Louth also found
in her study that tables of measurement, as such,
are seldom used in their entirety. Why then, should
a table, as such, be memorized?
3. The Hoyt-Peet Arithmetics retain foreign exchange.
It will be recalled that the Fourth Yearbook of
the Department of Superintendence recommended that
no provision for formal drill be made for work
in foreip-n exchange.
c
115
4. The greatest emphasis in each series of texts is
placad on reductions. The Fourth Yearbook of the
Department of Superintendence asserted that "no
reductions, either ascsndinj? or descending, occur
in business situations,"^ This has been confirmed
by the studies of Louth, Sala, and Traniello.
5. There is a modern tendency to eliminate computation
with compound denominate numbers entirely. But a large
number of examoles and problems with compound denominate
numbers have still been retained in these texts.
6. Tho much of the work with denominate numbers is done
incident?.lly in connection with other phases of
arithmetic, such work must be justified on the basis
of its social usefulness.
7. More emphasis needs to be placed on teaching the
use and meaning of units of wei,g;ht through commodities.
But to say that "fruits, grains, vegetables are
measured by the nint, the quart, the peck, and the
bushel" has little value. Each commodity must be
joined with the specific unit of measure with which
it is used.
1, Fourth Yearboo'.-., Department of Superintendence, p. 178,
cc
... t '
116
Selected bibliography
1. Fourth Yearbook, Department of Superintendence, ^'^ational
Education Association, 1926. pp.l73-220»
2. Lennes, N.J. The Teaching of 'Arithmetic. New York:
Macmillan Company, 1929, pp.278-288»
3. Louth, ^iary. iJnits of Measurement in Industry. B.U.
Graduate school thesis, 1931. ms.94p.
4. Moore, Vernette B, An Analysis of Denominate Numbers
with Data on Their Teaching. State ^niversity of Iowa,
Master's thesis of the Graduate College, 1929.
5. Sala, V. Denominate ^^umbers used in the Factories of
New Britain. B.U. School of Education thesis, 1931.
ms» 80p»
6. Traniello, A. Use of Tables of ^'Measurement in Business.
B.U. School of Education thesis, 1930. ms.70p.
7. Wilson, G.M. "Arithmetic", Department of Superintendence,
Third Yearbook, Washington, D.C. National Education
Association, 1925, p. 35-109,
8. Wilson, G.M. A Survey of the Social and business Usage
of Arithmetic. Contributions to Education, No. 100,
Teachers College, Columbia University, 1919,
9. Wilson, G.M. What Arithmetic Shall We Teach Houghton
Mifflin Compansi, 1926.
c
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